MS Seminar (Mathematics - String Theory)

Speaker: Alexander Schmitt(Freie Universität Berlin)
Title: Relative geometric invariant theory
Date (JST): Tue, Jul 16, 2019, 15:30 - 17:00
Place: Seminar Room A
Related File: 2361.pdf
Abstract: Classical invariant theory deals with polynomial invariants of algebraic forms, such as the discriminant of a quadratic form. Geometrically, it can be interpreted as the classification of hypersurfaces in projective space up to homogeneous changes coordinates. Hilbert proved that the ring ofthese invariants is a finitely generated algebra and Mumford turned invariant theory into a powerful theory for solving classification
problems in algebraic geometry. Algebraic forms fall naturally into two classes, the semistable forms and the nullforms. In applications, it is a key step to find the semistable forms or points with the help of the so-called Hilbert-Mumford criterion. Classically, the Hilbert-Mumford criterion works for projective and affine varieties. In more recent work, relative versions for projective morphisms have been discussed.

In the first part of the talk, we will review the basic formalism and results of geometric invariant theory. In the second part, we will speak about the relative setting. We will explain how a result of Reichstein implies the Hilbert-Mumford criterion of Gulbrandsen, Halle, and Hulek and why the Hilbert-Mumford criterion does not work in general. If time
permits, we will discuss the construction of the relative moduli space of vector bundles or principal bundles over the moduli space of stable curves as an application of relative geometric invariant theory. (This has some
connections to mathematical physics.)